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A note on non‐negative continuous time processes

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  • Henghsiu Tsai
  • K. S. Chan

Abstract

Summary. Recently there has been much work on developing models that are suitable for analysing the volatility of a continuous time process. One general approach is to define a volatility process as the convolution of a kernel with a non‐decreasing Lévy process, which is non‐negative if the kernel is non‐negative. Within the framework of time continuous autoregressive moving average (CARMA) processes, we derive a necessary and sufficient condition for the kernel to be non‐negative. This condition is in terms of the Laplace transform of the CARMA kernel, which has a simple form. We discuss some useful consequences of this result and delineate the parametric region of stationarity and non‐negative kernel for some lower order CARMA models.

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  • Henghsiu Tsai & K. S. Chan, 2005. "A note on non‐negative continuous time processes," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(4), pages 589-597, September.
    • Handle: RePEc:bla:jorssb:v:67:y:2005:i:4:p:589-597
    DOI: 10.1111/j.1467-9868.2005.00517.x
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    Cited by:

    1. Chunsheng Ma, 2017. "Vector Stochastic Processes with Pólya-Type Correlation Structure," International Statistical Review, International Statistical Institute, vol. 85(2), pages 340-354, August.
    2. P. Brockwell, 2014. "Recent results in the theory and applications of CARMA processes," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 66(4), pages 647-685, August.
    3. Iacus, Stefano M. & Mercuri, Lorenzo & Rroji, Edit, 2017. "COGARCH(p, q): Simulation and Inference with the yuima Package," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 80(i04).
    4. Creal, Drew D., 2008. "Analysis of filtering and smoothing algorithms for Lévy-driven stochastic volatility models," Computational Statistics & Data Analysis, Elsevier, vol. 52(6), pages 2863-2876, February.
    5. Fred Espen Benth & Jūratė Šaltytė Benth & Steen Koekebakker, 2008. "Stochastic Modeling of Electricity and Related Markets," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 6811, September.
    6. Fred Espen Benth & Jūratė Šaltytė Benth & Steen Koekebakker, 2007. "Putting a Price on Temperature," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 34(4), pages 746-767, December.
    7. Mercuri, Lorenzo & Perchiazzo, Andrea & Rroji, Edit, 2024. "A Hawkes model with CARMA(p,q) intensity," Insurance: Mathematics and Economics, Elsevier, vol. 116(C), pages 1-26.
    8. Benth, Fred Espen & Karbach, Sven, 2023. "Multivariate continuous-time autoregressive moving-average processes on cones," Stochastic Processes and their Applications, Elsevier, vol. 162(C), pages 299-337.

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